The celebrated estimates of Cwikel [1]} are a landmark application of trace-ideal techniques in mathematical physics. They assert that if \(f\in L_p(\mathbb {R}^n)\)  and \(g\)  is in the weak \(L_p\) -space \(L_{p,\infty }(\mathbb {R}^n)\)  with \(p>2\) , then the operator \(f(X)g(-i\nabla )\)  on \(L_2(\mathbb {R}^n)\)  is in the weak Schatten class \({L}_{p,\infty }\) , and we have
\(\big \Vert  f(X)g(-i\nabla )\big \Vert _{{L}_{p,\infty }}\le c_{np}\big \Vert f\big \Vert _{L_p} \big \Vert g\big \Vert _{L_{p,\infty }},\) 
